Cohomology group of an algebra

Question

I tried to guess what is a cohomology group of an algebra. I would like to find the correct definition of this. I know what is a cohomology group of a group, but I don't know how connect the second cohomology group to the central extension of a Lie-algebra. Could someone help me to tell the definition or explain to me this? Thanks!

Answer 1

The notion that you probably want is that of Hochschild (co)homology. This is a way of getting (co)homology groups for associative algebras, but it actually specializes to group cohomology and Lie algebra cohomology. The reason this is possible is because modules over a finite group $G$ are the same thing as modules over the group algebra $k[G]$, and similarly, modules over a Lie algebra $\mathfrak g$ are the same thing as modules over the universal enveloping algebra $\mathcal U\mathfrak g$. It turns out that (co)homology groups really only care about the category of modules, and so the framework of associative algebras subsumes that of finite groups and modules or lie algebras and their modules.

Of course, while this general framework gives you a unifying picture for everything, don't forget that you might have intepretations of cohomology groups in special cases that don't generalize.